Chaotic Time Series Analysis

<table>Schematic illustration for the indices of the nearest neighbors in <svg height="7.3249998" id="M580" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 8.25 7.3249998" width="8.25" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-66.14)">
<text transform="matrix(1,0,0,-1,-71.95,66.19)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐱</tspan>
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<g transform="translate(72,-63.57)">
<text transform="matrix(1,0,0,-1,-71.95,66.19)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐲</tspan>
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</svg> state space, respectively, where the number <svg height="12.8875" id="M582" style="vertical-align:-1.76814pt" version="1.1" viewbox="0 0 22.262501 12.8875" width="22.262501" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-61.69)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="6.2515001" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="11.452748" y="0">3</tspan>
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</svg> denote the <svg height="12.975" id="M583" style="vertical-align:-4.77652pt" version="1.1" viewbox="0 0 20.2125 12.975" width="20.2125" xmlns="http://www.w3.org/2000/svg">
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<g transform="translate(72,-61.62)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑟</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.85,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
<tspan style="font-size: 8.75px; " x="4.3497438" y="0">,</tspan>
<tspan style="font-size: 8.75px; " x="6.537744" y="0">𝑗</tspan>
</text>
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</svg> and <svg height="14.975" id="M584" style="vertical-align:-1.76814pt" version="1.1" viewbox="0 0 30.950001 14.975" width="30.950001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.975)">
<g transform="translate(72,-60.02)">
<text transform="matrix(1,0,0,-1,-71.95,61.84)">
<tspan style="font-size: 12.50px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,-65.7,67.01)">
<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
</text>
<text transform="matrix(1,0,0,-1,-62.22,61.84)">
<tspan style="font-size: 12.50px; " x="0" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="5.2012482" y="0">3</tspan>
</text>
<text transform="matrix(1,0,0,-1,-50.76,67.01)">
<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
</text>
</g>
</g>
</svg> the <svg height="12.975" id="M585" style="vertical-align:-4.77652pt" version="1.1" viewbox="0 0 20.7125 12.975" width="20.7125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.975)">
<g transform="translate(72,-61.62)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑠</tspan>
</text>
<text transform="matrix(1,0,0,-1,-66.45,63.31)">
<tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan>
<tspan style="font-size: 8.75px; " x="4.3497438" y="0">,</tspan>
<tspan style="font-size: 8.75px; " x="6.537744" y="0">𝑗</tspan>
</text>
</g>
</g>
</svg>.</table>

Mathematical Problems in Engineering

Chaotic Time Series Analysis

Figure 7